Audio amp, some speakers, volume control if not on audio amp already.
Some tact switches preferably soft silicone like top
Some 3 way switches
A switch for the eject button used on wiikey.
A fan to keep system cool and a slim lined heatsink
If you want batteries, this is where the costs start to add up.
GC uses 12V and draws around 2A, so you would want batteries of higher voltage so the voltage drops take long enough for the full capacity to deplete.
14.8V is what you want. And probably around 5Ah capacity to get 2 hours. Then you also need a protection circuit with the batteries to protect over discharge and a smart charger to prevent over charge.
If you want the best efficiency on battery life and more space, you would want to use a switching regulator rather than the stock voltage regulator on the GC. These little circuits come from TI. Need ones that could output 1.7v, and 3.3V at 10A and 12V at 6A. The mentioned voltages, are the voltages the gamecube runs on. And the averages matter because, the higher the possible current output, the cooler the chips stay and last their lifetime.
If you don't want to read all of this, just head on over to modretro forums and modded by bacteria forums, these 2 are both flooded with info